Question: What is the largest product of two primes that have a sum of 60?
Explanation: To maximize the product of the two numbers, we want the numbers to be as close to each other as possible. 30 and 30 are not prime, so we let one prime be in the 20s and one prime be in the 30s. The primes in those ranges are \[23, 29, 31, 37.\] We see that $29+31=60$ and are closest together, so the desired product is \[29 \cdot 31 = (30-1)(30+1)=30^2-1^2=900-1=\boxed{899}.\]